 in which \(\tau >0\) , \(V(x)\)  is a polynomial with positive even degree and a positive leading coefficient and \(dA\)  denotes Lebesgue measure on the space of \(n\times n\)  complex Hermitian matrices. It bears worth mentioning that setting \(\tau =n/2\)  and \(V(x)=x^2\)  gives rise to the Gaussian unitary ensemble introduced by Dyson [1]}, [2]}. The main result of [3]} establishes that the linear eigenvalue statistic \(\sum _{k} f(\lambda _k)\)  converges in distribution to a normal random variable with mean zero. The sums we consider in this paper can be thought of as randomized graph-theoretic versions of the sums originally considered by Johansson.
